The bottleneck in the proving algorithm of most elliptic-curve-based SNARK proof systems is Multi-Scalar-Multiplication (MSM) algorithm. In this paper we give an overview a variant Pippenger MSM together with set optimizations tailored for curves that admit twisted Edwards form. We prove case SNARK-friendly chains and cycles elliptic curves, which are useful recursive constructions. Our contrib...

Snarks are bridgeless cubic graphs with chromatic index χ = 4. A snark G is called critical if χ(G − {v, w}) = 3, for any two adjacent vertices v and w. For any k ≥ 2 we construct cyclically 5-edge connected critical snarks G having an independent set I of at least k vertices such that χ(G − I) = 4. For k = 2 this solves a problem of Nedela and Škoviera [6].

We determine the exact values of the circular chromatic index of the Goldberg snarks, and of a related family, the twisted Goldberg snarks.

Snarks are cubic graphs with chromatic index 0 = 4. A snark G is called critical if 0 (G?fv; wg) = 3 for any two adjacent vertices v and w, and it is called bicritical if 0 (G ? fv; wg) = 3 for any two vertices v and w. We construct innnite families of critical snarks which are not bicritical. This solves a problem stated by Nedela and Skoviera in 7].

Church's thesis claims that all effecticely calculable functions are recursive. A shortcoming of the various definitions of recursive functions lies in the fact that it is not a matter of a syntactical check to find out if an entity gives rise to a function. Eight new ideas for a precise setup of arithmetical logic and its metalanguage give the proper environment for the construction of a speci...

We develop a theory of factorisation of snarks — cubic graphs with edge-chromatic number 4 — based on the classical concept of the dot product. Our main concern are irreducible snarks, those where the removal of every nontrivial edge-cut yields a 3-edge-colourable graph. We show that if an irreducible snark can be expressed as a dot product of two smaller snarks, then both of them are irreducib...

Polyhedral embeddings of cubic graphs by means of certain operations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Grünbaum that no snark admits an orientable polyhedral embedding. This conjecture is verified for all snarks having up to 30 vertices using computer. On the other hand, for every nonorie...